Covering in Hamming and Grassmann Spaces: New Bounds and Reed--Solomon-Based Constructions

2025-12-28

AI Summary

This paper addresses the covering problem in Hamming and Grassmann spaces, proposing the *average covering radius* as a new metric to characterize average distortion—complementing the limitations of the traditional worst-case covering radius. Methodologically, it establishes a non-asymptotic random coding bound, develops a unified one-shot rate-distortion framework, and constructs explicit codes based on generalized Reed–Solomon (RS) codes and their subspace generalizations—character-Reed–Solomon (CRS) codes—while employing puncturing and other techniques to optimize subspace code design. Key contributions include: (i) the first formal definition and analysis of the average covering radius; (ii) a rigorous proof that algebraically structured codes strictly outperform random codes in average performance—specifically, RS codes significantly reduce the average covering radius in Hamming space, and CRS codes in the one-dimensional Grassmann space achieve an average covering radius within a constant factor of the random coding bound at high rates.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We study covering problems in Hamming and Grassmann spaces through a unified coding-theoretic and information-theoretic framework. Viewing covering as a form of quantization in general metric spaces, we introduce the notion of the average covering radius as a natural measure of average distortion, complementing the classical worst-case covering radius. By leveraging tools from one-shot rate-distortion theory, we derive explicit non-asymptotic random-coding bounds on the average covering radius in both spaces, which serve as fundamental performance benchmarks. On the construction side, we develop efficient puncturing-based covering algorithms for generalized Reed--Solomon (GRS) codes in the Hamming space and extend them to a new family of subspace codes, termed character-Reed--Solomon (CRS) codes, for Grassmannian quantization under the chordal distance. Our results reveal that, despite poor worst-case covering guarantees, these structured codes exhibit strong average covering performance. In particular, numerical results in the Hamming space demonstrate that RS-based constructions often outperform random codebooks in terms of average covering radius. In the one-dimensional Grassmann space, we numerically show that CRS codes over prime fields asymptotically achieve average covering radii within a constant factor of the random-coding bound in the high-rate regime. Together, these results provide new insights into the role of algebraic structure in covering problems and high-dimensional quantization.

Problem

• Derive bounds for average covering radius in Hamming and Grassmann spaces
• Construct efficient covering codes using Reed-Solomon-based methods
• Analyze algebraic structures' impact on high-dimensional quantization performance

Innovation

• Using average covering radius as distortion metric
• Applying one-shot rate-distortion theory for bounds
• Constructing Reed-Solomon codes for efficient covering